🏅 Olympiad maths · session 3 of 49 · 60 minutes

Counting without listing: the multiplication principle

How many outfits, codes or numbers can you make? Learn to count them without writing every one: list in order, draw a tree, then multiply the choices.

🖨 Practice sheet

A complete lesson · 60 minutes

Teach it, see it, practise it, all on this page.

Open this page with your learner and work down it, or press Teach this lesson to go through it one step at a time on a big screen. Every section has something to do.

See the teacher's plan

Warm-up4 min
Organised lists8 min
Trees9 min
Multiply7 min
Rules7 min
Activity 17 min
Activity 27 min
Activity 36 min
Quiz4 min
Set the project1 min

By the end you can

Make an organised list so nothing is missed or repeated
Draw a tree diagram for choices made one after another
Use the multiplication principle
Count when there are rules (no repeats, must be even, no zero first)

Before you start

Multiplication facts
Place value: hundreds, tens and ones

Have ready

Paper and pencil
Three coloured pencils

Classes 4 to 7 · ages 9 to 13

Warm-up · think first

You have 2 T-shirts (red, blue) and 3 pairs of shorts (black, grey, white). How many different outfits can you wear? Try to list them all.

Show the answer

6 outfits: red with each of the 3 shorts, and blue with each of the 3 shorts. 2 × 3 = 6. Today you will learn why multiplying works, and how it saves time when there are hundreds of outfits.

Teacher tip: Let them list freely first. Then ask: how can we be sure we did not miss one? That leads to listing in order.

Part 1 of 4 · learn, see, practise

Make an organised list

When you list possibilities, fix the first choice and go through every second choice, then move to the next first choice. This stops you missing or repeating any.

Arrangements of things in a line are a good example: write them in alphabetical order.

Example 1

In how many ways can three friends A, B and C stand in a line?

Start with A first: ABC, ACB.
Then B first: BAC, BCA.
Then C first: CAB, CBA.
That is 3 groups of 2.

Answer: 6 ways

Think first: List all the 2-digit numbers you can make from the digits 1, 2 and 5 without repeating a digit.

Practise it

Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.

Warm-up1In how many ways can the letters of CAT be arranged?

Warm-up2How many 2-digit numbers can be made from 3, 5, 7 without repeating a digit?

Core3Ana, Bo and Cy sit on 3 chairs in a row and Ana must sit in the middle. How many ways?

Stretch 🔥4How many different arrangements of the letters of BOOK are there?

Answers (for the teacher)

Part 2 of 4 · learn, see, practise

Tree diagrams

A tree diagram draws each choice as a branch. Every path from the start to the end of a branch is one possibility.

Trees are perfect when choices happen one after another: first coin toss, second coin toss; first course, main course.

Example 1

A coin is tossed twice. What are all the possible results?

The first toss splits into H and T.
Each of those splits again into H and T.
The four paths are HH, HT, TH and TT.

Answer: 4 results

Think first: Draw the tree for tossing a coin three times. How many paths are there?

Practise it

Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.

Warm-up1A coin is tossed and a die is rolled. How many results?

Core2A coin is tossed 4 times. How many results?

Core3Lunch: rice or roti, then dal, paneer or chole, then curd or salad. How many lunches?

Stretch 🔥4A coin is tossed 3 times. How many results have exactly 2 heads?

Answers (for the teacher)

Part 3 of 4 · learn, see, practise

Multiply the choices

Look at any tree: every branch at one level splits into the same number of branches at the next. So the number of paths is the product of the number of choices at each step.

This is the multiplication principle: if one choice can be made in a ways and the next in b ways, together they can be made in a × b ways (and so on for more steps).

Example 1

3 shirts, 2 trousers and 2 pairs of shoes. How many outfits?

Choose a shirt: 3 ways.
For each, choose trousers: 2 ways, so 3 × 2 = 6 so far.
For each of those, choose shoes: 2 ways, so 6 × 2 = 12.

Answer: 12 outfits

The outfit machine

Change how many shirts, trousers and shoes there are. The machine lists every outfit (when there are not too many) and multiplies.

Make exactly 12 outfits.
Make 24 outfits with at least 2 of everything.
Make 30 outfits.

Think first: A café has 4 sandwiches, 3 drinks and 2 desserts. How many meals of one of each?

Practise it

Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.

Warm-up15 tops and 3 skirts: how many outfits?

Core2How many 4-digit PINs (0000 to 9999) are there?

Core3Number plates have 2 letters (A to Z) then 2 digits. How many plates?

Stretch 🔥44 shirts, 3 trousers and 2 pairs of shoes, but the red shirt cannot be worn with the green trousers. How many outfits?

Answers (for the teacher)

15
10000
67600
22

Part 4 of 4 · learn, see, practise

Counting with rules

When there are rules, fill the places one by one and count the choices for each place, starting with the place that has a rule.

A 3-digit number cannot start with 0. If digits cannot repeat, each place has one choice fewer than the one before.

Example 1

How many 3-digit numbers have all different digits?

Hundreds: any digit except 0: 9 choices.
Tens: any digit except the one used: 9 choices (0 is allowed now).
Ones: 8 choices left.
9 × 9 × 8 = 648.

Answer: 648

Example 2

How many 2-digit numbers are even?

Tens digit: 1 to 9, so 9 choices.
Ones digit: 0, 2, 4, 6 or 8, so 5 choices.
9 × 5 = 45.

Answer: 45

Think first: How many 3-digit numbers have no zero in them at all?

Practise it

Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.

Warm-up1How many 3-digit numbers have all digits odd?

Core2How many 4-digit numbers have all different digits?

Core3How many 3-digit numbers are multiples of 5?

Stretch 🔥4How many 3-digit even numbers have all different digits?

Answers (for the teacher)

125
4536
180
328

Watch out

Common mistakes

Adding instead of multiplying

3 shirts and 4 trousers make 12 outfits, not 7. Add only when you choose one thing OR another.

Forgetting the zero rule

A number like 045 is really a 2-digit number. The first digit of a 3-digit number cannot be 0.

Counting repeats twice

If AB and BA are the same (like a pair of friends), the count is halved. Read whether order matters.

Class activity 1 of 3

How many ways?

Type each count and press Check.

14 flavours, 3 toppings, cone or cup: how many ice creams with one of each?

22-digit numbers from the digits 1, 2, 3, 4 if digits may repeat

32-digit numbers from the digits 1, 2, 3, 4 if digits may not repeat

4Ways for 4 children to stand in a line

53-digit numbers with no digit 0

6Ways to choose a captain and a vice-captain from 6 players

Answers (for the teacher)

Class activity 2 of 3

Paths on a grid

You walk on the lines of a grid from the top-left corner to the bottom-right corner, only moving right or down. Count the paths.

Draw a 1 by 1 square and count: 2 paths.
Draw a 2 by 2 grid. At each corner write the number of ways to reach it: it is the sum of the corner above and the corner to the left.
Fill bigger grids the same way.

1Paths across a 2 by 2 grid

2Paths across a 2 by 3 grid

3Paths across a 3 by 3 grid

4Paths across a 4 by 4 grid

Answers (for the teacher)

Class activity 3 of 3

Design a number plate

A country is designing number plates. Count how many plates each design gives.

Count the choices for each place.
Multiply.
When letters cannot repeat, each place has one choice fewer.

13 letters (A to Z), repeats allowed

23 letters without repeats

33 letters with a vowel first (A, E, I, O, U), repeats allowed

42 letters then 3 digits, repeats allowed

Answers (for the teacher)

17576
15600
3380
676000

Quiz · 8 questions

Check what you learned

Pick an answer to see if it is right and why. Your best score is saved on this device.

13 shirts and 4 trousers. How many outfits?

Show the answer

12. 3 × 4 = 12.

2How many 3-digit numbers use each of 1, 2 and 3 exactly once?

Show the answer

6. 3 × 2 × 1 = 6.

3How many 2-digit numbers are there?

Show the answer

90. 9 choices for tens × 10 for ones = 90 (10 to 99).

4A coin is tossed 3 times. How many results are possible?

Show the answer

8. 2 × 2 × 2 = 8.

5A president and a vice-president are chosen from 5 people. In how many ways?

Show the answer

20. 5 choices, then 4: 20.

6How many 3-letter codes can be made from A, B, C, D if letters may repeat?

Show the answer

64. 4 × 4 × 4 = 64.

7How many 2-digit numbers are even?

Show the answer

45. 9 tens digits × 5 even ones digits.

8There are 3 roads from A to B and 4 roads from B to C. How many ways to go from A to C through B?

Show the answer

12. 3 × 4 = 12.

After class · project

Design a menu

Design the menu for your own café or ice-cream stall, then count how many different orders customers can make.

Choose 3 or 4 categories (for example base, filling, sauce, drink).
Put 2 to 5 items in each category.
Work out how many orders have one item from each category.
Add one rule (for example: the chocolate sauce only goes with ice cream) and count again.
Draw a tree for a small part of your menu to show the counting.

What to make: A menu page with your categories, the total number of orders, one rule and its new count, and a small tree diagram.

Stretch: Count the orders where the customer may skip one category.

How it is marked

Look for	Done well when
Menu	At least three categories with items
Counting	The total uses multiplication correctly
Rule	The count with the rule is right and explained
Tree	A clear tree for part of the menu

Project template

Copy it onto paper or print the practice sheet.

Category 1: ______ items: ___________________ (count: ___)
Category 2: ______ items: ___________________ (count: ___)
Category 3: ______ items: ___________________ (count: ___)
Orders with one of each: ___ × ___ × ___ = ___
My rule: ____________________   Orders now: ___ (working: ________)

Homework

Practise at home

Try each on paper, then tap to see the answer and working. Or print the practice sheet.

How many ways can 5 books stand in a row on a shelf?
5 × 4 × 3 × 2 × 1 = 120
How many 3-digit numbers can be made from the digits 2, 4, 6, 8 if digits may repeat?
4 × 4 × 4 = 64
How many odd 3-digit numbers are there?
9 × 10 × 5 = 450
A lock has 3 dials each showing 0 to 9. How many codes are there?
10 × 10 × 10 = 1000

For parents and teachers

The 60-minute lesson plan

Use it as it is, or stretch any part over two sessions. Each row links to the part of the page you need.

Time	Part	The teacher	The learner
0–4 min	Warm-up	Ask the warm-up question and wait for a guess.	Guess, then reason.
4–12 min	Organised lists	Teach, show the examples step by step, then the practice.	Practise up to the stretch question.
12–21 min	Trees	Teach, use the interactive part, then the practice.	Practise; try the stretch.
21–28 min	Multiply	Teach and show the examples.	Practise.
28–35 min	Rules	Teach and show the examples.	Practise; the stretch can be homework.
35–42 min	Activity 1	Let the learner work alone; hint only after a wrong check.	Solve and check.
42–49 min	Activity 2	Follow the steps together for the first item.	Finish the investigation.
49–55 min	Activity 3	Set as homework if time is short.	Solve the puzzles.
55–59 min	Quiz	Run the quiz and review mistakes.	Answer and read why.
59–60 min	Set the project	Show the template and the due date.	Start the project.

Questions to ask

Why do we multiply and not add?
Why does the hundreds place have only 9 choices?
When does the order of choosing not matter?

If it feels hard

Use real objects (pencils of 3 colours) and draw every tree in full before multiplying.

If it feels easy

How many ways can 6 friends sit at a round table if only the order around the table matters?

FAQ

Counting without listing: the multiplication principle: questions

What is the multiplication principle in counting?

If one choice can be made in a ways and the next in b ways, the two together can be made in a × b ways. It extends to any number of steps.

How many 3-digit numbers have all different digits?

9 × 9 × 8 = 648: 9 choices for the first digit (not 0), then 9, then 8.

When should I use a tree diagram?

When choices happen one after another and you want to see every possibility. It also shows why the multiplication principle works.

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